Question

triangle xyz was transformed by ( d_{o,3} ). what is the length of side xy?
○ 1.5
○ 4.5
○ 3
○ 18

Explanation:

Step1: Understand the Dilation

The transformation \( D_{O, 3} \) means a dilation with center \( O \) and scale factor 3. This implies that the image (the larger triangle) is 3 times the size of the pre - image (the smaller triangle \( XYZ \)).

Step2: Find the Length of \( XY \)

We know that the length of \( X'Y' \) is 9. Since dilation with a scale factor of 3 means that \( \text{Length of } X'Y'=3\times\text{Length of } XY \). Let the length of \( XY \) be \( x \). Then we have the equation \( 9 = 3x \).
To solve for \( x \), we divide both sides of the equation by 3: \( x=\frac{9}{3}=3 \)? Wait, no, wait. Wait, maybe I mixed up the pre - image and the image. Wait, actually, the smaller triangle is the pre - image and the larger one is the image. Wait, looking at the diagram, the length from \( O \) to \( Y \) and \( O \) to \( Y' \), or the sides. Wait, another way: the side \( X'Y' \) is the image of \( XY \) under dilation with scale factor 3. So \( X'Y' = 3\times XY \). We know \( X'Y'=9 \), so \( XY=\frac{X'Y'}{3}=\frac{9}{3} = 3 \)? Wait, no, wait the options: 1.5, 4.5, 3, 18. Wait, maybe I got the scale factor reversed. Wait, maybe the scale factor is \( \frac{1}{3} \) for the pre - image? Wait, no, \( D_{O,3} \) means that the image is 3 times the pre - image. Wait, let's check the sides. The side \( ZZ' \): the length from \( Z \) to \( Z' \): the length of \( OZ \) is 4? Wait, no, the length of \( OZ \) is 4? Wait, the length of \( OZ' \): wait, the length of \( ZZ' \): no, the length of \( OZ \) is 4? Wait, the length of \( OZ \) is 4, and \( OZ' \): wait, the length from \( O \) to \( Z \) is 4? Wait, no, the diagram shows that the length of \( ZZ' \) related? Wait, no, let's look at the side \( XZ \) and \( X'Z' \). The length of \( X'Z' \) is 12, and the length of \( XZ \): let's see, the length from \( O \) to \( Z \) is 4? Wait, no, the length of \( OZ \) is 4? Wait, the length of \( OZ \) is 4, and \( OZ' \): wait, the length of \( X'Z' \) is 12, and \( XZ \): if the scale factor is 3, then \( X'Z' = 3\times XZ \), so \( XZ=\frac{12}{3}=4 \)? No, that doesn't match. Wait, maybe the pre - image is the larger triangle? No, dilation notation \( D_{O,k} \): if \( k>1 \), it's an enlargement, if \( 0 < k < 1 \), it's a reduction. Wait, the green triangle is the pre - image, the blue one is the image. So \( D_{O,3} \) means the image (blue) is 3 times the pre - image (green). So the side \( X'Y' \) (blue) is 3 times \( XY \) (green). So \( X'Y' = 3\times XY \), so \( XY=\frac{X'Y'}{3}=\frac{9}{3}=3 \)? But wait, let's check another side. The side \( XZ \) (green) and \( X'Z' \) (blue). \( X'Z' = 12 \), so \( XZ=\frac{12}{3}=4 \)? But in the diagram, the length from \( O \) to \( Z \) is 4? Wait, no, the length of \( OZ \) is 4, and \( OZ' \): wait, the length of \( OZ \) is 4, and \( OZ' \) is \( OZ + ZZ' \)? No, dilation is about the center. So the distance from \( O \) to \( Y \) and \( O \) to \( Y' \): if \( Y' \) is the image of \( Y \), then \( OY'=3\times OY \). Let's say \( OY = y \), then \( OY' = 3y \). The length of \( Y'Y \) would be \( OY' - OY=2y \), but maybe that's not helpful. Wait, going back to the side \( X'Y' = 9 \), and we need to find \( XY \). Since dilation with scale factor 3, \( XY=\frac{X'Y'}{3}=\frac{9}{3}=3 \). Wait, but let's check the options. 3 is one of the options. Wait, but maybe I made a mistake. Wait, maybe the scale factor is \( \frac{1}{3} \). Wait, if the pre - image is the larger triangle and the image is the smaller one, then \( XY = 3\times X'Y' \)? No, that would be 27, which is not an option. Wait, no, the problem says "Triangle XYZ was transformed by \( D_{O,3} \)" to get the larger triangle? Wait, no, the green triangle is XYZ, and the blue one is X'Y'Z'. So \( D_{O,3} \) transforms XYZ to X'Y'Z', so X'Y'Z' is the image, so scale factor 3, so X'Y' = 3*XY. So 9 = 3*XY, so XY = 3. Wait, but let's check the other sides. X'Z' = 12, so XZ = 12/3 = 4. And OZ: the length from O to Z is 4? Wait, the length from O to Z is 4, and OZ' would be 12? No, the diagram shows that the length from O to Z is 4? Wait, no, the length of OZ is 4, and OZ' is 12? Then 12 is 3 times 4, which matches the scale factor 3. So that makes sense. So then XY: X'Y' is 9, so XY is 9/3 = 3.

Answer:

3